JPS autumn 2010 T. Umeda (Hiroshima) 1

ウィルソンクォークを用いた ウィルソンクォークを用いた

　　 NNff=2+1 QCD =2+1 QCD の状態方程式の研の状態方程式の研

究究

Takashi Umeda (Hiroshima Univ.)Takashi Umeda (Hiroshima Univ.)

for WHOT-QCD Collaborationfor WHOT-QCD Collaboration

JPS meeting, Kyushu-koudai, Fukuoka, 13 Sep. 2010JPS meeting, Kyushu-koudai, Fukuoka, 13 Sep. 2010

/12

JPS autumn 2010 2

Quark Gluon Plasma in Lattice QCD

from the Phenix group web-site

Observables in Lattice QCDObservables in Lattice QCD

Phase diagram in (T, Phase diagram in (T, μμ, m, mudud, m, mss))

Transition temperatureTransition temperature

Equation of state ( ε/TEquation of state ( ε/T44, p/T, p/T44,...),...)

Hadronic excitationsHadronic excitations

Transport coefficientsTransport coefficients

Finite chemical potentialFinite chemical potential

etc...etc...http://www.gsi.de/fair/experiments/

T. Umeda (Hiroshima) /12

JPS autumn 2010 T. Umeda (Hiroshima) 3

Choice of quark actions on the lattice

Most (T, μ≠0) studies done with staggerd-type quarksMost (T, μ≠0) studies done with staggerd-type quarks less computational costsless computational costs a part of chiral sym. preserved ...a part of chiral sym. preserved ...

NNff=2+1, almost physical quark mass, (μ≠0)=2+1, almost physical quark mass, (μ≠0)

4th-root trick to remove unphysical “tastes”4th-root trick to remove unphysical “tastes”

non-locality “Validity is not guaranteed”non-locality “Validity is not guaranteed”

It is important to cross-check with It is important to cross-check with

theoretically sound lattice quarks like Wilson-type quarkstheoretically sound lattice quarks like Wilson-type quarks

Our aim is to investigateOur aim is to investigate

QCD Thermodynamics with Wilson-type quarksQCD Thermodynamics with Wilson-type quarks

WHOT-QCD CollaborationWHOT-QCD Collaboration

/12

JPS autumn 2010 T. Umeda (Hiroshima) 4

Fixed scale approach to study QCD thermodynamics

Temperature Temperature T=1/(NT=1/(Ntta)a) is varied by is varied by NNtt at fixed at fixed aa

AdvantagesAdvantages

- Line of Constant Physics- Line of Constant Physics

- T=0 subtraction for renorm.- T=0 subtraction for renorm.

(spectrum study at T=0 )(spectrum study at T=0 )

- larger 1/a in whole T region- larger 1/a in whole T region

DisadvantagesDisadvantages

- T resolution by integer N- T resolution by integer Ntt

- UV cutoff eff. at high T- UV cutoff eff. at high T

a : lattice spacinga : lattice spacingNNtt : lattice size in temporal direction : lattice size in temporal direction

Temperatures in each approachTemperatures in each approach

/12

fine ~

coarse

fine ~

coarse

fixed N

fixed N tt appro

ach

approach

fixed scale approachfixed scale approach

JPS autumn 2010 T. Umeda (Hiroshima) 5

T=0 & T>0 configurations for Nf=2+1 QCD

T=0 simulation: on 28T=0 simulation: on 2833 x 56 x 56 by CP-PACS/JLQCDby CP-PACS/JLQCD Phys. Rev. D78 (2008) 011502Phys. Rev. D78 (2008) 011502

- - RG-improved Iwasaki glue + NP-improved Wilson quarksRG-improved Iwasaki glue + NP-improved Wilson quarks

- β=2.05, κ- β=2.05, κudud=0.1356, κ=0.1356, κss=0.1351=0.1351

- V~(2 fm)- V~(2 fm)33 , a=0.07 fm, , a=0.07 fm,

- configurations available on the - configurations available on the ILDG/JLDGILDG/JLDG

T>0 simulations: on 32T>0 simulations: on 3233 x N x Ntt (N (Ntt=4, 6, ..., 14, 16) lattices=4, 6, ..., 14, 16) lattices

RHMC algorithm, same parameters as T=0 simulationRHMC algorithm, same parameters as T=0 simulation

/12

JPS autumn 2010 T. Umeda (Hiroshima) 6

Formulation for Nf=2+1 improved Wilson quarks

Noise methodNoise method ( #noise = 1 for each color & spin indices ) ( #noise = 1 for each color & spin indices )

Phys. Rev. D73, 034501Phys. Rev. D73, 034501CP-PACS/JLQCDCP-PACS/JLQCD

/12

JPS autumn 2010 T. Umeda (Hiroshima) 7

Beta-functions from CP-PACS/JLQCD results

Trace anomaly needsTrace anomaly needs Beta-functions Beta-functions in N in Nff=2+1 QCD=2+1 QCD

Direct fit method Direct fit method Phys. Rev. D64 (2001) 074510Phys. Rev. D64 (2001) 074510

fit fit ββ,,κκudud,,κκss as functions of as functions of

with fixed with fixed

/12

JPS autumn 2010 T. Umeda (Hiroshima) 8

Beta-functions from CP-PACS/JLQCD results

χχ22/dof=5.3/dof=5.3χχ22/dof=1.6/dof=1.6 χχ22/dof=2.1/dof=2.1

Meson spectrum by CP-PACS/JLQCD Meson spectrum by CP-PACS/JLQCD Phys. Rev. D78 (2008) 011502Phys. Rev. D78 (2008) 011502. .

fit fit ββ,,κκudud,,κκss as functions of as functions of

3 (β) x 5 (κ3 (β) x 5 (κudud) x 2 (κ) x 2 (κss) = 30 data points) = 30 data points

only statistical erroronly statistical error

/12

JPS autumn 2010 T. Umeda (Hiroshima) 9

Trace anomaly in Nf=2+1 QCD

Nt config. (x 5MD traj.)Nt config. (x 5MD traj.) SSgg S Sq q

(**)(**)

56 56 13001300(*)(*) 980 980 16 1542 64716 1542 647 14 1448 647 14 1448 647 12 1492 69512 1492 695 10 863 48710 863 487 8 628 5208 628 520 6 657 3606 657 360 4 802 2954 802 295

(*) T=0 (Nt=56) by CP-PACS/JLQCD(*) T=0 (Nt=56) by CP-PACS/JLQCD SSgg calculated with 6500traj. calculated with 6500traj.(**) thermal. = 1000 traj.(**) thermal. = 1000 traj.

Preliminary Preliminary

/12

JPS autumn 2010 T. Umeda (Hiroshima) 10

Trace anomaly in Nf=2+1 QCD

HotQ

CD

, H

otQ

CD

, arX

iv:1

005

.1131

arX

iv:1

005

.1131

HISQHISQ

stoutstout

peak height = 4~6peak height = 4~6 in recent Staggered resultsin recent Staggered results ( m( mq q ~ m~ mqq

phys. phys. ))

Preliminary Preliminary

/12

S. B

orsa

nyi e

t al.,

S. B

orsa

nyi e

t al.,

arX

iv:1

007

.2580

arX

iv:1

007

.2580

WHOT-QCD CollaborationWHOT-QCD Collaboration

JPS autumn 2010 T. Umeda (Hiroshima) 11

Equation of State in Nf=2+1 QCD

PreliminaryPreliminary T-integrationT-integration

is performed by the trapezoidal is performed by the trapezoidal rule (straight line interpolation). rule (straight line interpolation).

ε/Tε/T44 is calculated from is calculated from

Large error in whole T regionLarge error in whole T region

SB limitSB limit

/12

JPS autumn 2010 T. Umeda (Hiroshima) 12

Summary & outlook

Equation of stateEquation of state

More statistics More statistics are needed in the lower temperature region are needed in the lower temperature region

Results at different scales (β=1.90 by CP-PACS/JLQCD)Results at different scales (β=1.90 by CP-PACS/JLQCD)

NNff=2+1 QCD just at the physical point=2+1 QCD just at the physical point

the physical point (pion mass ~ 140MeV) by PACS-CSthe physical point (pion mass ~ 140MeV) by PACS-CS

β=1.90 is appropriate to control stat. error at lower T.β=1.90 is appropriate to control stat. error at lower T.

Odd Nt config. generation is necessary.Odd Nt config. generation is necessary.

Finite densityFinite density

Taylor expansion method to explore EOS at Taylor expansion method to explore EOS at μμ≠≠00

We presented the EOS in NWe presented the EOS in Nff=2+1 QCD using improve Wilson quarks=2+1 QCD using improve Wilson quarks

/12

JPS autumn 2010 T. Umeda (Hiroshima) 13

T-integration method to calculate the EOS

We propose a new method (“We propose a new method (“T-integration methodT-integration method”) ”)

to calculate the EOS at fixed scalesto calculate the EOS at fixed scales

Our method is based on Our method is based on the trace anomaly (interaction measure),the trace anomaly (interaction measure),

and and the thermodynamic relation.the thermodynamic relation.

T.Umeda et al. (WHOT-QCD), Phys.Rev.D79 (2009) 051501(R)T.Umeda et al. (WHOT-QCD), Phys.Rev.D79 (2009) 051501(R)

JPS autumn 2010 T. Umeda (Hiroshima) 14

Test in quenched QCD

Our results are roughlyOur results are roughly consistent with previous results. consistent with previous results.

Our results deviate from the Our results deviate from the fixed Nfixed Ntt=8 results [*] =8 results [*] at higher T ( aT~0.3 or higher )at higher T ( aT~0.3 or higher )

Trace anomaly is sensitive toTrace anomaly is sensitive to spatial volume at lower Tspatial volume at lower T (below T(below Tcc).). V > (2fm)V > (2fm)33 is ncessarry. is ncessarry.

~~

[*] [*] G. Boyd et al., NPB469, 419 (1996) G. Boyd et al., NPB469, 419 (1996)

JPS autumn 2010 T. Umeda (Hiroshima) 15

Quark mass dependence of Trace anomaly

HotQCD arXiv1005.1131HotQCD arXiv1005.1131

0.05m0.05mss

0.2m0.2mss

HotQCD PRD91,054504(2010)HotQCD PRD91,054504(2010) CP-PACS CP-PACS PRD64,074510 (2001)PRD64,074510 (2001)

peak height of the Trace anomalypeak height of the Trace anomaly small quark mass dependencesmall quark mass dependence

Our result seems to be reasonable !Our result seems to be reasonable ! but small mbut small mqq is necessary. is necessary.